Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. This relies on another contribution of this paper, which eliminates the need to examine a factorial number of permutations of simplices with the same value. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude.

翻译：利用持续同调指导优化已成为拓扑数据分析的新兴应用。现有方法将持续性计算视为黑箱，仅将梯度反向传播至特定配对涉及的单纯形上。我们展示了如何利用持续性计算中的循环和链结构，将梯度赋给域中更大的子集。特别地，我们证明在作为通用损失函数构建基础的特定情形中，该问题可在线性时间内精确求解。这一结论依赖于本文的另一贡献——无需检验具有相同值的单纯形阶乘数量级的排列。实验表明，我们的算法具有实际优势：优化所需步数减少了一个数量级。